Suppose that a deep shaft were drilled in Earth\'s crust near one of the poles,

Question

Suppose that a deep shaft were drilled in Earth's crust near one of the poles, where the surface temperature is -48°C, to a depth where the temperature is 590°C. (a) What is the theoretical limit to the efficiency of an engine operating between these temperatures? (b) If all the energy released as heat into the low-temperature reservoir were used to melt ice that was initially at -48°C, at what rate could liquid water at 0°C be produced by a 76.8 MW power plant (treat it as an engine)? The specific heat of ice is 2220 J/kg·K; water's heat of fusion is 333 kJ/kg. (Note that the engine can operate only between 0°C and 590°C in this case. Energy exhausted at -48°C cannot be used to raise the temperature of anything above -48°C.)

Explanation / Answer

a)
effeciency = 1 - TL/TH
= 1- (-48 + 273)/(590+273)
= 1- (225/863)
= 0.74

b)
maximum power released = 0.74 * 76.8 MW = 56.832 MW = 5.6832*10^7 J/s

Let mass if ice melted per s be m Kg

Q = m*L + m*C*(delta T)
= m*333000 + m*2220*(48)
= m*439560

Q = 5.6832*10^7
m*439560 = 5.6832*10^7
m = 129.3 Kg

Answer: 129 Kg/s

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